What happens in the case of a missing parent or an abandonment?
This depends on the law in the state where you are seeking adoption. States usually have statutes permitting adoption without the consent of a missing parent but only after diligent efforts are made to find the parent. This usually requires the assistance of a lawyer to prevent an adoption from being later upset should the missing parent reappear.
Abandonment by the missing parent and failure to support or communicate with the child generally gives you grounds to adopt on proper notice to the parent(s) who abandoned the child. Assistance from an adoption attorney is strongly advised.